A $p$-specific spectral multiplier theorem with sharp regularity bound for Grushin operators
Lars Niedorf
TL;DR
The paper resolves the p-specific $L^p$-spectral multiplier problem for the Grushin operator $L$ on $\mathbb{R}^{d_1}\times\mathbb{R}^{d_2}$ by achieving the sharp regularity threshold $s>(d_1+d_2)(1/p-1/2)$. It employs a spectrum-truncated restriction framework grounded in the joint functional calculus of $L$ and $T=(-\Delta_y)^{1/2}$ to reduce the analysis to spectral slices where the geometry is effectively Euclidean up to a scale factor. This yields $L^p$-boundedness of $F(\sqrt L)$ under $\|F\|_{\mathrm{sloc},s}$ for $1\le p\le p_{d_1,d_2}$ and also proves Bochner–Riesz means boundedness in the same regime. By avoiding weighted restriction estimates and exploiting the Grushin geometry, the work sharpens the understanding of sub-Laplacians in degenerate, Grushin-type settings and provides near-optimal regularity requirements for p-specific multiplier problems.
Abstract
In a recent work, P. Chen and E. M. Ouhabaz proved a $p$-specific $L^p$-spectral multiplier theorem for the Grushin operator acting on $\mathbb{R}^{d_1}\times\mathbb{R}^{d_2}$ which is given by \[ L =-\sum_{j=1}^{d_1} \partial_{x_j}^2 - \bigg( \sum_{j=1}^{d_1} |x_j|^2\bigg) \sum_{k=1}^{d_2}\partial_{y_k}^2. \] Their approach yields an $L^p$-spectral multiplier theorem within the range $1< p\le \min\{ \frac{2d_1}{d_1+2},\frac{2(d_2+1)}{d_2+3} \}$ under a regularity condition on the multiplier which is sharp only when $d_1\ge d_2$. In this paper, we improve on this result by proving $L^p$-boundedness under the expected sharp regularity condition $s>(d_1+d_2)(1/p-1/2)$. Our approach avoids the usage of weighted restriction type estimates which played a key role in the work of P. Chen and E. M. Ouhabaz, and is rather based on a careful analysis of the underlying sub-Riemannian geometry and restriction type estimates where the multiplier is truncated along the spectrum.